Properties

Label 2-334620-1.1-c1-0-43
Degree $2$
Conductor $334620$
Sign $-1$
Analytic cond. $2671.95$
Root an. cond. $51.6909$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 3·7-s − 11-s − 4·19-s + 25-s + 6·29-s − 7·31-s − 3·35-s + 2·37-s + 2·41-s + 43-s − 10·47-s + 2·49-s + 6·53-s + 55-s + 6·59-s − 3·61-s − 7·67-s + 4·71-s + 15·73-s − 3·77-s − 5·79-s − 6·83-s + 10·89-s + 4·95-s − 19·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s − 0.301·11-s − 0.917·19-s + 1/5·25-s + 1.11·29-s − 1.25·31-s − 0.507·35-s + 0.328·37-s + 0.312·41-s + 0.152·43-s − 1.45·47-s + 2/7·49-s + 0.824·53-s + 0.134·55-s + 0.781·59-s − 0.384·61-s − 0.855·67-s + 0.474·71-s + 1.75·73-s − 0.341·77-s − 0.562·79-s − 0.658·83-s + 1.05·89-s + 0.410·95-s − 1.92·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(334620\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(2671.95\)
Root analytic conductor: \(51.6909\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 334620,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 \)
good7 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.73006530954735, −12.37108208154993, −11.85565723977778, −11.42714614243169, −10.94361056983941, −10.76336000972501, −10.13554815505607, −9.675807349534689, −9.044967430009166, −8.580540271670595, −8.218787437607193, −7.835842356978875, −7.364361293278399, −6.782471603804403, −6.387980010695007, −5.683025720797776, −5.252822202107569, −4.717492154844924, −4.358189708887614, −3.805015997571344, −3.221526994588276, −2.542443977974905, −2.033822338199802, −1.463485689882956, −0.7646784629936721, 0, 0.7646784629936721, 1.463485689882956, 2.033822338199802, 2.542443977974905, 3.221526994588276, 3.805015997571344, 4.358189708887614, 4.717492154844924, 5.252822202107569, 5.683025720797776, 6.387980010695007, 6.782471603804403, 7.364361293278399, 7.835842356978875, 8.218787437607193, 8.580540271670595, 9.044967430009166, 9.675807349534689, 10.13554815505607, 10.76336000972501, 10.94361056983941, 11.42714614243169, 11.85565723977778, 12.37108208154993, 12.73006530954735

Graph of the $Z$-function along the critical line